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%I #35 Feb 02 2022 23:51:07
%S 5,13,17,37,53,61,73,89,97,109,113,137,149,157,173,193,197,233,257,
%T 269,277,293,313,317,337,353,373,389,397,421,433,457,557,577,593,613,
%U 617,653,661,673,677,701,733,757,761,773,797,821,829,853,857,877,937,953
%N Odd primes p with 4 zeros in any period of the Fibonacci numbers mod p.
%C Also, primes that do not divide any Lucas number. - _T. D. Noe_, Jul 25 2003
%C Although every prime divides some Fibonacci number, this is not true for the Lucas numbers. In fact, exactly 1/3 of all primes do not divide any Lucas number. See Lagarias and Moree for more details. The Lucas numbers separate the primes into three disjoint sets: (A053028) primes that do not divide any Lucas number, (A053027) primes that divide Lucas numbers of even index and (A053032) primes that divide Lucas numbers of odd index. - _T. D. Noe_, Jul 25 2003; revised by _N. J. A. Sloane_, Feb 21 2004
%H T. D. Noe, <a href="/A053028/b053028.txt">Table of n, a(n) for n=1..1000</a>
%H C. Ballot and M. Elia, <a href="http://www.fq.math.ca/Papers1/45-1/quartballot01_2007.pdf">Rank and period of primes in the Fibonacci sequence; a trichotomy</a>, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B2).
%H Nicholas Bragman and Eric Rowland, <a href="https://arxiv.org/abs/2202.00704">Limiting density of the Fibonacci sequence modulo powers of p</a>, arXiv:2202.00704 [math.NT], 2022.
%H J. C. Lagarias, <a href="http://projecteuclid.org/euclid.pjm/1102706452">The set of primes dividing the Lucas numbers has density 2/3</a>, Pacific J. Math., 118. No. 2, (1985), 449-461.
%H J. C. Lagarias, <a href="http://projecteuclid.org/euclid.pjm/1102622818">Errata to: The set of primes dividing the Lucas numbers has density 2/3</a>, Pacific J. Math., 162, No. 2, (1994), 393-396.
%H Diego Marques and Pavel Trojovsky, <a href="https://doi.org/10.2478/tmmp-2014-0019">The order of appearance of the product of five consecutive Lucas numbers</a>, Tatra Mountains Math. Publ. 59 (2014), 65-77.
%H Pieter Moree, <a href="http://msp.org/pjm/1998/186-2/p03.xhtml">Counting Divisors of Lucas Numbers</a>, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
%H M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">Fibonacci sequence modulo m</a>
%H H. Sedaghat, <a href="https://www.fq.math.ca/Papers1/52-1/Sedaghat.pdf">Zero-Avoiding Solutions of the Fibonacci Recurrence Modulo A Prime</a>, Fibonacci Quart. 52 (2014), no. 1, 39-45. See p. 45.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>
%F A prime p = prime(i) is in this sequence if p > 2 and A001602(i) is odd. - _T. D. Noe_, Jul 25 2003
%t Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; badP={}; Do[p=Prime[n]; k=1; While[k<p&&Mod[Lucas[k], p]>0, k++ ]; If[k==p, AppendTo[badP, p]], {n, 200}]; badP
%Y Cf. A001176.
%Y Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)), A053027, A053032.
%K nonn
%O 1,1
%A _Henry Bottomley_, Feb 23 2000
%E Edited: Name clarified. Moree and Renault link updated. Ballot and Elia reference linked. - _Wolfdieter Lang_, Jan 20 2015
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